CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedaudited 2026-09-14

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A stable Serre diagonal need not split its abutment

Statement

Assume the Axiom of Choice and fix $n \geq 2$ . Let $q : B Z \to B C_{n}$ induce the quotient $Z ↠ C_{n}$ , and let $F_{q} ⟶ E_{q} \overset{p_{q}}{\to} B C_{n}$ be its mapping-path fibration. Then $E_{q} ≃ B Z$ , the fiber is connected, and, under the fiber-inclusion identification, $π_{1} (F_{q}) = n Z \subseteq Z = π_{1} (E_{q}), π_{k} (F_{q}) = 0 (k > 1) .$ The total-degree-one stable Serre pieces are $E_{0, 1}^{\infty} = n Z ≅ Z, E_{1, 0}^{\infty} = Z / n,$ but $H_{1} (E_{q}; Z) = Z$ has the nonsplit filtration $0 \subset n Z \subset Z$ . Thus knowing the stable Serre diagonal does not split the abutment.

Facts & Assumptions

Given: AC, $n \geq 2$ , the quotient-induced based map $q$ , and integral coefficients.

[A1]

The Axiom of Choice is assumed exactly for the classifying-space models in [F1].

[F1]

The classifying space of a discrete group is a K(G,1) identifies $B Z$ and $B C_{n}$ as connected CW $K (Z, 1)$ and $K (C_{n}, 1)$ models under AC.

[F2]

Mapping path factorization factors $q = p_{q} j_{q}$ , with $j_{q}$ a homotopy equivalence and $p_{q}$ a Hurewicz fibration.

[F3]

Long exact sequence of homotopy groups of a fibration supplies the group and component exact sequence of this mapping-path fibration.

[F4]

The first Hurewicz map is abelianization computes and compares first homology of the connected fiber and total space.

[F5]

Homological Serre spectral sequence supplies the finite total-degree-one filtration and its stable associated-graded pieces. Serre edge maps come from projection and fiber inclusion identifies the first filtration subgroup with the image of fiber inclusion.

Verification

technique · compute the homotopy fiber, then read the actual first Serre filtration before testing whether its extension splits

1.1

Since $j_{q}$ is a homotopy equivalence, [F1, F2] identify $π_{1} (E_{q})$ with $Z$ , all its higher homotopy groups with zero, and $(p_{q})_{*}$ with the quotient $Z ↠ C_{n}$ . The component end of [F3] is $Z ↠ C_{n} ⟶ π_{0} (F_{q}) ⟶ * .$ The first map is onto, so exactness gives one fiber component. The group part then gives an injection $π_{1} (F_{q}) ↪ Z$ with image the kernel $n Z$ . For $k > 1$ , the adjacent homotopy groups of both $E_{q}$ and $B C_{n}$ vanish, so [F3] gives $π_{k} (F_{q}) = 0$ .

A1F1F2F3

2.1

All three spaces used here are connected. Their fundamental groups $n Z$ , $Z$ , and $C_{n}$ are abelian, so [F4] identifies their first homology groups with those same groups. Naturality of Hurewicz identifies the fiber-inclusion map on first homology with $n Z ↪ Z .$ By [F5], its image is precisely the first Serre filtration term $F_{0} H_{1} (E_{q}) = n Z$ . Strong convergence in total degree one therefore gives $E_{0, 1}^{\infty} = F_{0} H_{1} (E_{q}) = n Z, E_{1, 0}^{\infty} = H_{1} (E_{q}) / F_{0} H_{1} (E_{q}) = Z / n .$ This uses the actual edge image, so it does not assume that either stable term already splits off.

F4F5step 1.1

3.1

If $0 \subset n Z \subset Z$ split, a section of $Z ↠ Z / n$ would embed a nonzero element of order $n$ in the torsion-free group $Z$ , which is impossible for $n \geq 2$ . Thus the abutment is not the direct sum $n Z \oplus Z / n$ , even though these are its two stable pieces. The witness is explicit, not merely an appeal to the existence of extension problems.

step 2.1

4.1

The fiber and total space are nonempty and connected. The zero higher homotopy groups, zero and first filtration terms, both total-degree-one stable positions, both maps in the short exact filtration, and the first permitted case $n = 2$ were checked. The excluded value $n = 1$ would give a trivial quotient and no nonsplit extension. AC is used only by [F1]; mapping paths, the homotopy exact sequence, Hurewicz, Serre filtration, and the finite torsion argument add no choice. No converse is asserted.

A1F1F2F3F4F5step 1.1step 2.1step 3.1∎

Source notes

Hatcher's extension warning, printed p. 527, uses the nonsplit sequence $0 \to Z \to Z \to Z / n \to 0$ to warn that stable terms need not sum to the abutment. Steps 1.1–3.1 realize exactly that extension as an actual Serre filtration.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

48 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Hatcher, Algebraic Topology, extension warning before Example 5.4 (standard reference, not scraped)